Technical report TR-LSR-2015-03
Chair of Automatic Control Engineering
Technische Unversität München
March 2015
Detailed derivations for the analysis of the fundamental dynamics based
leader-follower structures*
Philine Donner1,2 , Franz Christange1 and Martin Buss1,2
Abstract— In this report, the transfer functions used in [1]
that describe the behavior of the two different leader-follower
structures in interaction with the fundamental dynamics are
derived in detail.
d
θE
aL
d
2γL
Kd
B̂L
B
2
I. F LOW IMITATION APPROACH
ϑ̇r
leader
A. Reference input transfer function G
fi
ϑ̇ˆr
Figure 1 shows the block diagram of the leader-follower
structure with flow imitation approach and ideal linear fundamental dynamics. Rearrangement of the block diagram in
Fig. 1 leads to the block diagram displayed in Fig. 2. The
highlighted intermediate transfer function Gfi1 is
Gfi1
=
1−
1
s
s
1
B
γ
s F B̂F Tf s+1
s2
+
( T1f
γLd Kd B̂B s + γLd Kd B̂B
−
L
d B 1
γF
B̂F Tf
+
.
(1)
L
γLd Kd B̂B )s
L
1
Tf
+
γLd Kd B̂B T1f
L
B
2
fundamental dynamics
s
Tf s+1
d
θE
Gfi1
γLd Kd B̂B
L
ϑ̇r
1
s
d B
γF
B̂
F
ϑr
s
Tf s+1
(2)
Fig. 2. Rearranged block diagram of the flow imitation control structure
d (s).
for the computation of the transfer function Gfi (s): ϑr (s) = Gfi (s)θE
According to the final value theorem, the system energy
ϑr approaches
1 d (2) d
= θE ,
ϑr (t → ∞) = lim sGfi (s) θE
s→0
s
Fig. 1. Block diagram of the control structure with flow imitation follower
and fundamental dynamics.
.
B. Stationary transfer behavior
(3)
C. Relative follower contribution γF
For the computation of the relative follower contribution γF , consider the block diagram rearrangement in Fig. 3.
From Fig. 3 with
d
as t → ∞ for a step of height θE
in the reference
d
d
variable θE (t) = σ(t)θE .
*This work was supported in part within the ERC Advanced Grant
SHRINE Agreement No. 267877 (www.shrine-project.eu) and in part by the
Technische Universität München - Institute for Advanced Study (www.tumias.de), funded by the German Excellence Initiative
1 Chair
of
Automatic
Control
Engineering,
Technische
Universität München, Theresienstr. 90, 80333 München, Germany,
{philine.donner, franz.christange, mb}@tum.de
2 TUM
ϑr
follower
Based on (1) the reference input transfer function ϑr (s) =
d
(s) of a leader interacting with a flow imitation
Gfi (s)θE
follower is computed to
Gfi =
aF
d
2γF
B̂F
1
s
Institute for Advanced Study, Technische Universität München,
Lichtenbergstrasse 2a, 85748 Garching, Germany
Gfi2 =
1
1−
1
d B
γF
B̂F Tf s+1
.
(4)
we can compute the transfer function which yields the
amount of energy the leader contributed ϑr,L (s) based on
d
the reference input θE
(s)
GfiL =
γLd Kd B̂B (s +
d B
s2 + ( T1f − γF
B̂
F
L
1
Tf
+
1
d B 1
Tf − γF B̂F Tf )
γLd Kd B̂B )s + γLd Kd B̂B T1f
L
L
.
(5)
d
θE
1
s
γL Kd B̂B
L
Gfi2
ϑr,L
leader
ϑr
d
θE
s
γF B̂B Tf s+1
F
1
s
ˆ
ϑ̇r,L
d
θE
L
ϑr
ϑr,L
1
s
γL Kd B̂B
ϑr
1
s
B̂F
2
d
θ̂E
1
d)
K̂d (1−γF
ˆ
ϑ̇r
B
2
ϑ̇r
ˆ
ϑ̇r,F
1
Tf s+1
aL
d
2γL
Kd
B̂L
s
Tf s+1
kγ
aF
d
2γF
K̂d
B̂F
B
2
fundamental dynamics
1
Tf s+1
follower
Gfi2
Fig. 3. Rearranged block diagram of the flow imitation control structure
for the computation of the energy contributed by the leader ϑr,L =
d
Gfi
L (s)θE (s).
Fig. 4. Block diagram of the control structure with goal estimation follower
and fundamental dynamics.
the intermediate transfer function Gge
2 results to
γd
Based on the final value theorem the relative leader
contribution γL can be defined
1 d !
d
ϑr,L (t → ∞) = lim sGfiL (s) θE
= γL θ E
,
s→0
s
(6)
which yields for the flow imitation approach
d
γLfi = lim GfiL (s) = 1 − γF
s→0
B
B̂F
F
Tf s + 1 + kγ 1−γ
d
Gge
2 =
F
,
d
γF
d B ( 1
Tf s2 + (1 + kγ 1−γ
−
k
γ
γ F B̂
d
d − Tf K̂d ))s
1−γ
F
F
F
(10)
and the reference input transfer function is
γLd Kd B̂B s + γLd Kd B̂B
Gge =
,
L
s2 + ( T1f +
(7)
d
kγ γF
d
Tf 1−γF
L
(1 −
1
Tf
B
B̂F
γd
F
(1 + kγ 1−γ
d )
F
d
) + kγ γF
K̂d B̂B +
+ γLd Kd B̂B )s + γLd Kd B̂B
L
L
1
Tf
and consequently a relative follower contribution
fi
d
γF
= 1 − γLfi = γF
B
B̂F
.
(8)
D. Stability
From the system denominator we know that according to the Routh-Hurwitz criterion, the linear system is
d B 1
+ γLd B̂B Kd ) > 0
asymptotically stable if ( T1f − γF
B̂F Tf
L
and γLd Kd B̂B T1f > 0. The latter is always fulfilled, as
L
γLd , Kd , B, B̂L , Tf > 0.
II. G OAL ESTIMATION
=
1 + kγ
d K̂
2γF
d
B̂F
1
1
d ) Tf s+1 2
B̂F K̂d (1−γF
C. Relative follower contribution γF
For the computation of the relative follower contribution,
we proceed similarly as for the flow imitation approach. The
rearrangement steps shown in Fig. 6 build upon Fig. 5. With
the intermediate transfer function
F
Tf s + 1 + kγ 1−γ
F
=
γd
d B ( 1
F
Tf s + 1 + kγ 1−γ
d − kγ γF
d − Tf K̂d )
B̂F 1−γF
F
(13)
the transfer function from goal energy to energy contributed
d
by the leader ϑr,L (s) = Gge
L (s)θE (s) results in
Gge
L
=
d
γL
Kd B̂B (s +
L
2
s +
d
2γF
K̂d
B̂F
B. Stationary transfer behavior
Application of the final value theorem shows that the
d
system energy ϑr approaches the goal energy θE
1 d (11) d
ϑr (t → ∞) = lim sGge (s) θE
= θE ,
(12)
s→0
s
d
d
(t) =
in the reference variable θE
for a step of height θE
d
σ(t)θE .
Gge
3 (s)
The block diagram for the fundamental dynamics based
leader-follower structure with goal estimation follower is
given in Fig. 4. Figure 5 shows the rearrangement steps of the
original block diagram in Fig. 4 for the purpose of computing
d
the reference input transfer function ϑr (s) = Gge (s)θE
(s).
With
kγ
γd
γd
A. Reference input transfer function Gge
Gge
1
F
F
(1 + kγ 1−γ
d )
F
(11)
,
(9)
( T1f
+
+
1
Tf
+
d
kγ γ F
Tf 1−γ d
F
d
kγ γF
Tf 1−γ d
F
d
γL
Kd B̂B
L
(1 −
)s +
(1 −
B
B̂F
B
B̂F
)+
d
) + kγ γ F
K̂d B̂B )
F
d
kγ γF
K̂d B̂B
d
γL
Kd B̂B T1f
L
F
(1 +
+
d
γF
kγ 1−γ
d
F
)
(14)
d
θE
d
θE
Gge
1
aL
d
2γL
Kd
B̂L
L
B
2
ϑ̇r
aF
d
2γF
K̂d
B̂F
Gge
3
ϑr
d
θE
1
Tf s+1
+
Gge
2
(
B
Gge
1 2
ϑr,L
−1
ϑ̇r
γLd Kd B̂B
L
Tf s+1
1
s
γLd Kd B̂B
ϑr
d
θE
1
−Tf )s
K̂d (1−γ d )
F
B
2
L
s
1
d ) Tf s+1
K̂d (1−γF
(
1
s
ϑr
1
s
B̂F
1
1
d ) Tf s+1 2
K̂d (1−γF
kγ
1
s
γLd Kd B̂B
ϑr,L
ϑr
1
s
1
−Tf )s
K̂d (1−γ d )
F
Tf s+1
Gge
3
Fig. 6. Rearranged block diagram of the goal estimation control structure
for the computation of the energy contributed by the leader ϑr,L =
d
Gge
L (s)θE (s).
ge
Gge
. The reference input transfer function Gge is
d G
ϑr →θ̂E
given in (11). To compute the intermediate transfer function
d
Gge
d from system energy ϑr to estimated goal energy θ̂E
ϑr →θ̂E
we extract
B
Gge
1 2
Fig. 5.
Rearranged block diagrams of the goal estimation control
structure for the computation of the transfer function Gge (s): ϑr (s) =
d (s).
Gge (s)θE
d
θ̂E
=
1
K̂d (1 −
d)
γF
ˆ
ϑ̇r,L +
1
ϑr
Tf s + 1
(17)
and
Similar as done for the flow imitation approach, we
compute the relative leader contribution γL based on the final
value theorem (see (6))
γLge
= lim
s→0
GfiL (s)
=1−
1
d B
kγ γF
( 1−γ
d − Tf K̂d )
B̂
F
F
γd
, (15)
F
1 + kγ 1−γ
d
ˆ
ϑ̇r,L =
d
K̂d d
s
1
B̂F kγ 2γF
ϑr −
(θ̂E −ϑr ) (18)
Tf s + 1
Tf s + 1 2
B̂F
from Fig. 41 . Insertion of (18) into (17) and some rearrangements yield the intermediate transfer function
F
and equivalently a relative follower contribution
ge
γF
= 1 − γLge =
d B
kγ γF
B̂F
1
( 1−γ
d
F
− Tf K̂d )
γd
Gge
=
ϑ →θ̂ d
r
.
(16)
F
1 + kγ 1−γ
d
F
!
ge
d
As described in [1], we can enforce γF
= γF
as long as
1
B = B̂F by choosing kγ = 1−T K̂ .
f
d
D. Stability
According to the characteristic polynomial (denominator
of (11) and (14)) the goal estimation approach is stable as
d
γF
k
B
d B
long as T1f + Tγf 1−γ
)+kγ γF
K̂d +γLd B̂B Kd > 0
d (1−
B̂
B̂
and γLd Kd B̂B
L
1
Tf
F
(1 + kγ
F
d
γF
d
1−γF
F
L
) > 0. The latter is the case as
long as we make sure that kγ > 0, i.e. Tf K̂d < 1, because
d
γLd , Kd , B, B̂L , Tf , γF
> 0.
d
E. Estimated goal energy θ̂E
The computation of the transfer function from goal end
d
ergy θE
to estimated goal energy θ̂E
is computed as Gge
=
θ̂ d
E
E
1
d )s
K̂d (1−γF
Tf s +
γd
F
+ kγ 1−γ
d + 1
d
γF
kγ 1−γ
d
F
F
.
(19)
+1
The final value theorem applied to (19) yields
ge
lims→0 Gge
= 1 (see (12)),
d = 1. Because lims→0 G
ϑr →θ̂E
the estimated goal energy converges to the actual goal
d
d
energy θ̂E
(t → ∞) = θE
.
III. G OAL ESTIMATION WITH SECOND ORDER FILTERS
In Sec. VI of [1] we show results for second-order filters
ge,o2
s
1
and Ghp
= T 2 s2 +2D
Gge,o2
= T 2 s2 +2D
lp
f Tf s+1
f Tf s+1
f
f
ge
1
instead of the first-order filters Gge
lp = Tf s+1 and Ghp =
s
Tf s+1 . The derivations are equivalent to the ones for the
original goal estimation approach as given in the previous
section. Therefore, we report on our results only.
ˆ
1 Note that (18) and (17) are in frequency domain and ϑ̇
r,L is used as a
variable name, but not to indicate an actual time derivative.
A. Reference input transfer function Gge,o2
γd
ge,o2
G
=
F
γLd Kd B̂B (Tf2 s2 + 2Df Tf s + 1 + kγ 1−γ
d )
L
with b = 2Df Tf +
c=1+
+
d
γF
kγ 1−γ
d
F
F
Tf2 s3 + bs2 + cs + d
d
kγ γF
K̂d B̂B
F
Tf2
R EFERENCES
γLd Kd B̂B Tf2 ,
L
+
1
d
K̂d (1−γF
d
γF
+ kγ 1−γ d
F
d
+ kγ γF
K̂d B̂B (2Df Tf −
γLd Kd B̂B 2Df Tf ,
L
F
and d =
γLd Kd B̂B (1
L
(20)
[1] P. Donner, F. Christange, and M. Buss, “Fundamental dynamics based
adaptive energy control for cooperative swinging of complex pendulumlike objects,” in IEEE Proc. CDC, 2015 (submitted).
)+
).
B. Stationary transfer behavior
Application of the final value theorem shows that the
d
system energy ϑr approaches the goal energy θE
1 d (20) d
ϑr (t → ∞) = lim sGge,o2 (s) θE
= θE ,
s→0
s
(21)
d
d
for a step of height θE
in the reference variable θE
(t) =
d
σ(t)θE .
C. Relative follower contribution γF
Gge,o2
L
γLd Kd B̂B (Tf2 s2 + es + f )
L
=
(22)
Tf2 s3 + bs2 + cs + d
with the same denominator as (20) and
d
K̂d B̂B Tf2 , and
e = 2Df Tf + kγ γF
F
γd
1
B
d
F
(
f = 1 + kγ 1−γ
d − kγ γF K̂d
d ) + 2Df Tf ). Based
B̂F K̂d (1−γF
F
on the final value theorem this leads to a relative follower
contribution
ge,o2
γF
=
d
kγ γF
K̂d B̂B ( K̂
F
1
d
d (1−γF
d
γF
)
.
(23)
1 + kγ 1−γ d
F
We can enforce the desired follower contribution if B = B̂F
ge,o2 !
d
, which yields
by choosing kγ such that γF
= γF
kγo2 =
1
1 − 2Df Tf K̂d
.
(24)
D. Stability
From the denominator of (20) we infer that the goal
estimation approach with second order filters is stable as
long as Tf , b, c, d > 0. This is always fulfilled for Tf and
the sums b and d, as long as kγ > 0, i.e. 2Df Tf K̂d < 1,
d
because γLd , Kd , K̂d , B, B̂L , B̂F , Df , Tf , γF
> 0.
d
E. Estimated goal energy θ̂E
The computation of the transfer function from goal end
d
ergy θE
to estimated goal energy θ̂E
is again computed
ge,o2
ge,o2
ge,o2
as Gθ̂d
= Gϑ →θ̂d G
. The reference input transfer
r
E
E
function Gge,o2 is given in (20). The intermediate transfer
function is
Gge,o2
=
ϑ →θ̂ d
r
E
1
d )s
K̂d (1−γF
Tf2 s2
The final value theorem applied to (25) yields
lims→0 Gge,o2
= 1. Thus, with lims→0 Gge,o2 = 1
d
ϑr →θ̂E
(see (21)), the estimated goal energy converges to the actual
d
d
goal energy θ̂E
(t → ∞) = θE
.
γd
F
+ kγ 1−γ
d + 1
F
γd
F
+ 2Df Tf s + kγ 1−γ
d + 1
F
.
(25)